Most wave phenomena encountered in physics textbooks consider planar or spherical waves, which both have in common that their wavefronts form separated planes in space. However, waves of different topology can theoretically exist. An interesting class of such waves are so-called vortex waves, which are also known as waves with a topological charge or waves with a phase singularity.
Vortex waves were first discovered in radio waves and later found many applications in light optics. Such waves carry an orbital angular momentum (OAM) of m h per primary particle, in which the topical charge m is a non-zero integer, e.g. +1 or −1. The orbital angular momentum depends on the spatial distribution of the electromagnetic field, i.e. is carried by the vorticity of the wave and distinct from the angular momentum attributable to polarization.
Vortex waves are used in applications such as quantum information, nano-manipulation and astrophysics. Vortex waves have been obtained from different types of waves, such as radio, acoustic and X-ray waves. Recently, also electron vortex beams have been produced. Electron waves are routinely used in transmission electron microscopes because of their advantageous short wavelength, e.g. of the order of picometers, for kinetic energies of a few hundred keV. This small wavelength of accelerated electrons makes them ideal candidates for creating vortex beams of atomic size. Furthermore, electrons are charged particles and therefore carry a magnetic moment of mμB per electron in addition to the orbital angular momentum of m h per electron. This connection of OAM to magnetism makes them ideal candidates to probe the magnetic state of materials they interact with. In combination with the small wavelengths that can be obtained, this may lead to atomic resolution magnetic mapping of materials.
One application of electron vortex beams may be found in the field of electron energy loss spectroscopy (EELS). EELS is a spectroscopic technique used in transmission electron microscopy to measure the energy loss of the fast electrons when scattering inelastically in a material. The energy loss spectrum contains information about the type of atoms in the material, their chemical bonding, the electronic state and their valency. An attractive feature of EELS is that it can be obtained with a spatial resolution below 1 Å. Atomic resolution EELS experiments have been performed that show atom by atom the constitution of a given material. This is particularly of interest near interfaces and defects in materials.
The available information in EELS may be expanded to include magnetic information by making use of vortex beams, because the conservation of total angular momentum may influence the dipole selection rules that govern the possible excitations in EELS. For example, for ferromagnetic Fe and Co, a spectrum can be obtained that is similar to what is commonly obtained from X ray magnetic chiral dichroism (XCMD). XMCD makes use of absorption differences in circularly polarised X rays, while EELS with vortex electron waves may create the same incoming angular momentum with an electron beam, e.g. an electron wave carrying m=1. However, electron beams have the advantage over X-rays that atomic resolution may be achievable, as is routinely demonstrated in transmission electron microscopy.
It should be noted that a technique that offers magnetic information was already available in EELS under the name of Energy Loss Magnetic Chiral Dichroism (EMCD). EMCD is based on the interference of Bragg scattered electron beams by the crystal combined with inelastic scattering. In a situation with well defined crystal orientation and thickness, a spectrum very close to XMCD could also be obtained. However, precise control over thickness and orientation limits the range of applications in which EMCD can be used. EMCD is furthermore fundamentally limited to a spatial resolution bigger than a few unit cells, e.g. 2 nm, because elastic diffraction is essential in creating the signal. The signal to noise ratio of the technique is furthermore relatively low. Vortex electron beams on the other hand may have no fundamental limit to the maximum spatial resolution, apart from the wavelength, the orientation of the crystal plays no important role because the interference is caused by the vorticity of the beam rather than by Bragg scattering and a substantially larger signal to noise ratio may be achievable. Methods are known in the art to produce electron vortex waves which use holographic reconstruction techniques. Such methods work by illuminating a computer calculated grating structure with a planar electron reference beam to obtain a wave with a predefined phase. The grating is typically cut from a thin metal foil, e.g. a thickness of a few 100 nm of Pt, by using a focused ion beam instrument (FIB). An example of such grating is illustrated in FIG. 1, in which a fork-shaped discontinuity can be seen that may be typical for such gratings. This is an easy method to reproduce, and in principle a grating for any value of m may be produced with this method. However, this method has the disadvantage that the grating simultaneously produces three output beams, as shown by the electron intensity as obtained from such grating in the far field depicted in FIG. 2: the vortex wave of interest, the reference beam and the complex conjugate of the vortex wave of interest, i.e. a vortex wave of opposite handedness. This means that the total electron current available is distributed over the three beams. Furthermore, the grating may typically only transmit about 50% of the electrons, which further reduces the available current in the vortex beam of interest, e.g. to a maximum of ⅛ of the total current. A sufficient current may for example be important for obtaining a high signal to noise ratio. Since these three beam components are simultaneously present, it may be difficult to isolate a signal coming from the vortex beam of interest. It may be possible to overcome this disadvantage with other apertures which select only the beam of interest, for example a spiral aperture may be used.
An alternative method known in the art to produce vortex electron beams is by making use of a phase grating, which is similar to a phase grating for photons, but for electrons the grating substrate has to be extremely thin to produce a phase shift of 2π, e.g. less than 100 nm. This means that contamination on such a grating may deteriorate its function over time as the phase will change without further actions being taken.